TS EAMCET · Maths · Binomial Theorem
If the expression \(5^{2 n}-48 n+\mathrm{k}\) is divisible by 24 for all \(n \in \mathbb{N}\), then the least positive integral value of \(k\) is
- A \(47\)
- B \(48\)
- C \(24\)
- D \(23\)
Answer & Solution
Correct Answer
(D) \(23\)
Step-by-step Solution
Detailed explanation
For \(n=1\), the expression is: \(5^{2(1)} - 48(1) + k = 25 - 48 + k = -23 + k\) Since the expression is divisible by 24: \(-23 + k \equiv 0 \pmod{24}\) \(k \equiv 23 \pmod{24}\) The least positive integral value of \(k\) is \(23\).
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